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When is statistical significance not significant?

Dalson Britto Figueiredo Filho

Political Science Department, Federal University of Pernambuco (UFPE), Brazil

Ranulfo Paranhos

Social Science Institute, Federal University of Alagoas (UFAL), Brazil

Enivaldo C. da Rocha

Political Science Department, Federal University of Pernambuco (UFPE), Brazil

Mariana Batista

Ph.D candidate in Political Science, Federal University of Pernambuco (UFPE), Brazil

Jos? Alexandre da Silva Jr.

Social Science School, Federal University of Goi?s (UFG), Brazil

Manoel L. Wanderley D. Santos

Department of Political Science, Federal University of Minas Gerais (UFMG), Brazil

Jacira Guiro Marino

Carlos Drummond de Andrade School (FCDA), Brazil

The article provides a non-technical introduction to the p value statistics. Its main purpose is to help researchers make sense of the appropriate role of the p value statistics in empirical political science research. On methodological grounds, we use replication, simulations and observational data to show when statistical significance is not significant. We argue that: (1) scholars must always graphically analyze their data before interpreting the p value; (2) it is pointless to estimate the p value for non-random samples; (3) the p value is highly affected by the sample size, and (4) it is pointless to estimate the p value when dealing with data on population.

Keywords: p value statistics; statistical significance; significance tests.

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When is statistical significance not significant?

The basic problem with the null hypothesis significance test in political science is that it often does not tell political scientists what they think it is telling them. (J. Gill)

The statistical difficulties arise more generally with findings that are suggestive but not statistically significant. (A. Gelman and D. Weakliem)

The research methodology literature in recent years has included a full frontal assault on statistical significance testing. (J. E. McLean and J. M. Ernest)

Statistical significance testing has involved more fantasy than fact. (R. Carver)

Introduction1

What is the fate of a research paper that does not find statistically significant results? According to Gerber, Green and Nickerson (2001: 01), "articles that do not reject the null hypothesis tend to go unpublished" Likewise, Sigelman (1999: 201) argues that "statistically significant results are achieved more frequently in published than unpublished studies. Such publication bias is generally seen as the consequence of a widespread prejudice against non significant results"2. Conversely, Henkel (1976: 07) argues that significance tests "are of little or no value in basic social science research, where basic research is identified as that which is directed toward the development and validation of theory". Similarly, McLean and Ernest (1998: 15) point out that significance tests provide no information about the practical significance of an event, or about whether or not the result is replicable. More directly, Carver (1978; 1993) argues that all forms of significance test should be abandoned3. Considering this controversy, what is the appropriate role of the p value statistic in empirical political science research? This is our research question. This paper provides a non-technical introduction to the p value statistic. Our main purpose is to help students in making sense of the appropriate role of the p value statistic in empirical political science research. On methodological grounds, we use observational data from the Quality of Government Institute4 simulations and replicate results from Anscombe (1973), Cohen (1988) and Hair et al., (2006) to show what can be learned from the p value statistic. There are situations where interpretation of the p value requires caution and we suggest four warnings: (1) scholars must always graphically analyze their data

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Dalson Britto Figueiredo Filho, Ranulfo Paranhos Enivaldo C. da Rocha, Mariana Batista Jos? Alexandre da Silva Jr., Manoel L. Wanderley D. Santos and Jacira Guiro Marino

before interpreting the p value; (2) it is pointless to estimate the p value for non-random

samples; (3) the p value is highly affected by the sample size, and (4) it is pointless to es-

timate the p value when dealing with data from population5.

The remainder of the paper consists of three sections. Firstly, we outline the under-

lying logic of null hypothesis significance tests, and we define what p value is and how it

should be properly interpreted. Next, we replicate Anscombe (1973), Cohen (1988) and

Hair et al., (2006) data, using basic simulation and analyze observational data to explain

our view regarding the proper role of the p value statistic. We close with a few concluding

remarks on statistical inference in political science.

What the p value is, what it means and what it does not

Statistical inference is based on the idea that it is possible to generalize results from a sample to the population6. How can we assure that relations observed in a sample are not simply due to chance? Significance tests are designed to offer an objective measure to inform decisions about the validity of the generalization. For example, one can find a negative relationship in a sample between education and corruption, but additional information is necessary to show that the result is not simply due to chance, but that it is "statistically significant". According to Henkel (1976), hypothesis testing is:

Employed to test some assumption (hypothesis) we have about the population against a sample from the population (...) the result of a significance test is a probability which we attach to a descriptive statistic calculated from a sample. This probability reflects how likely it is that the statistic could have come from a sample drawn from the population specified in the hypothesis (Henkel, 1976: 09)7.

In the standard approach to significance testing, one has a null hypothesis (Ho) and an alternative hypothesis (Ha), which describe opposite and mutually exclusive patterns regarding some phenomena8. Usually while the null hypothesis (Ho) denies the existence of a relationship between X and Y, the alternative hypothesis (Ha) supports that X and Y are associated. For example, in a study about the determinants of corruption, while the null hypothesis (Ho) states that there is no correlation between education and corruption, the alternative hypothesis (Ha) states that these variables are correlated, or more specifically indicates the direction of the relationship; that education and corruption are negatively associated9.

Usually, scholars are interested in rejecting the null hypothesis in favor of the alternative hypothesis, since the alternative hypothesis represents the corroboration of the theoretical expectations of the researcher. Also, as identified by Gerber, Green and Nickerson

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When is statistical significance not significant?

(2001), there is a publication bias that favors papers that successfully reject the null hypothesis. Therefore, scholars have both substantial and practical incentives to prefer statistically significant results.

McLean and Ernest (1998: 16) argue that "a null hypothesis (Ho) and an alternative hypothesis (Ha) are stated, and if the value of the test statistic falls in the rejection region the null hypothesis is rejected in favor of the alternative hypothesis. Otherwise the null hypothesis is retained on the basis that there is insufficient evidence to reject it". In essence, the main purpose of hypothesis test is to help the researcher to make a decision about two competing views of the reality. According to Henkel (1976),

Significance testing is assumed to offer an advantage over subjective evaluations of closeness in contexts such as that illustrated above where there are no specific criteria for what constitutes enough agreement (between our expectations and our observations) to allow us to continue to believe our hypothesis, or constitutes great enough divergence to lead us to suspect that our hypothesis is false. In a general sense, tests of significance, as one approach to assessing our beliefs or assumptions about reality, differ from the common sense approach only in the degree to which the criterion for closeness of, or correspondence between, observed and expected results are formalized, that is, specific and standardized across tests. Significance testing allows us to evaluate differences between what we expect on the basis of our hypothesis, and what we observe, but only in terms of one criterion, the probability that these differences could have occurred by `chance' (Henkel, 1976: 10).

In theory, the p value is a continuous measure of evidence, but in practice it is typically trichotomized approximately into highly significant, marginally significant, and not statistically significant at conventional levels, with cutoffs at p0.01, p0.05 and p>0.10 (Gelman, 2012: 2). According to Cramer and Howitt (2004),

The level at which the null hypothesis is rejected is usually set as 5 or fewer times out of 100. This means that such a difference or relationship is likely to occur by chance 5 or fewer times out of 100. This level is generally described as the proportion 0.05 and sometimes as the percentage 5%. The 0.05 probability level was historically an arbitrary choice but has been acceptable as a reasonable choice in most circumstances. If there is a reason to vary this level, it is acceptable to do so. So in circumstances where there might be very serious adverse consequences if the wrong decision were made about the hypothesis, then the significance level could be made more stringent at, say, 1% (Cramer and Howitt, 2004: 151).

Figure 1 illustrates the logic of null hypothesis significance testing.

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Dalson Britto Figueiredo Filho, Ranulfo Paranhos Enivaldo C. da Rocha, Mariana Batista Jos? Alexandre da Silva Jr., Manoel L. Wanderley D. Santos and Jacira Guiro Marino

Figure 1. Null Hypothesis Significance Testing illustrated

Source: Gill (1999) 10

We know that the area under the curve equates to 1 and can be represented by a probability density function. As we standardize the variable to a standard normal, we have a mean of zero and the spread is described by the standard deviation. Importantly, given that this curve's standard deviation equals 1, we know that 68.26% of all observations are between -1 and +1 standard deviation, 95.44% of all observations will fall between -2 and +2 standard deviation and 99.14% of all cases are between -3 and +3 standard deviation. The shaded area represents the probability of observing a result from a sample as extreme as we observed, assuming the null hypothesis in population is true. For example, in a regression of Y on X the first step is to state the competing hypotheses:

Ho: bx = 0 Ha: bx 0 While the null hypothesis states that the effect of X on Y is zero (bx = 0), the alternative hypothesis states that the effect is different from zero (bx 0). The second step is to compare our estimate with the parameters specified under the null hypothesis. The more our estimate approximates to the parameters specified by the null hypothesis, the less confidence we have in rejecting it. The more distant our estimate is from the parameters specified by the null hypothesis, the more confidence we have in rejecting Ho in favor of Ha. The p value statistic is a conditional probability, the probability of obtaining the observed or more extreme result given that the null hypothesis is true. To estimate the p value or the probability value, we should proceed as follows; (1) write down both the null (Ho) and the alternative hypothesis (Ha); (2) calculate the difference between the expected value under the null hypothesis and the observed value based on sample data; (3) standardize the difference into Z scores, and (4) estimate the probability of the alternative hypothesis assuming that the null hypothesis is true. Algebraically,

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When is statistical significance not significant?

Where represents the observed value, 0 represents the value under the null, represents the variance of the distribution and n represents the sample size (number of observations). When the difference between the observed value and the value under the null increases, all other things constant, higher is the Z. Similarly, when the sample size gets bigger, all other things constant, the variance is lower and the Z is higher. The Z score is higher, and the p value statistic is lower. Therefore, the p value depends not only upon the effect magnitude but is, by definition, determined by the sample size.

To make the estimation of the p value statistic more practical to political science scholars, we use observational data from the Quality of Government Institute11. The first step is to state both the null hypothesis and the alternative hypothesis.

Ho: r 0 Ha: r ................
................

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